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Graphs and Combinatorics

, Volume 35, Issue 3, pp 767–778 | Cite as

On Upper Total Domination Versus Upper Domination in Graphs

  • Enqiang Zhu
  • Chanjuan Liu
  • Fei DengEmail author
  • Yongsheng Rao
Original Paper
  • 21 Downloads

Abstract

A total dominating set of a graph G is a dominating set S of G such that the subgraph induced by S contains no isolated vertex, where a dominating set of G is a set of vertices of G such that each vertex in \(V(G){\setminus } S\) has a neighbor in S. A (total) dominating set S is said to be minimal if \(S{\setminus } \{v\}\) is not a (total) dominating set for every \(v\in S\). The upper total domination number \(\varGamma _t(G)\) and the upper domination number \(\varGamma (G)\) are the maximum cardinalities of a minimal total dominating set and a minimal dominating set of G, respectively. For every graph G without isolated vertices, it is known that \(\varGamma _t(G)\le 2\varGamma (G)\). The case in which \(\frac{\varGamma _t(G)}{\varGamma (G)}=2\) has been studied in Cyman et al. (Graphs Comb 34:261–276, 2018), which focused on the characterization of the connected cubic graphs and proposed one problem to be solved and two questions to be answered in terms of the value of \(\frac{\varGamma _t(G)}{\varGamma (G)}\). In this paper, we solve this problem, i.e., the characterization of the subcubic graphs G that satisfy \(\frac{\varGamma _t(G)}{\varGamma (G)}=2\), by constructing a class of subcubic graphs, which we call triangle-trees. Moreover, we show that the answers to the two questions are negative by constructing connected cubic graphs G that satisfy \(\frac{\varGamma _t(G)}{\varGamma (G)}>\frac{3}{2}\) and a class of regular non-complete graphs G that satisfy \(\frac{\varGamma _t(G)}{\varGamma (G)}=2\).

Keywords

Upper domination number Upper total domination number Subcubic graphs 

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (61872101, 61672051, 61702075), the China Postdoctoral Science Foundation under Grant (2017M611223).

References

  1. 1.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. Society for Industrial and Applied Mathematics, Philadelphia (1999)CrossRefzbMATHGoogle Scholar
  2. 2.
    Brešar, B., Henning, M.A., Rall, D.F.: Total domination in graphs. Discrete Math. 339, 1665–1676 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, J., He, K., Du, R., Zheng, M., Xiang, Y., Yuan, Q.: Dominating set and network coding-based routing in wireless mesh networks. IEEE Trans. Parallel Distrib. Syst. 26(2), 423–433 (2015)CrossRefGoogle Scholar
  4. 4.
    Cockayne, E.J., Dawes, R.M., Hedetniemi, S.T.: Total domination in graphs. Networks 10, 211–219 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cyman, J., Dettlaff, M., Henning, M.A., Lemańska, M., Raczek, J.: Total domination versus domination in cubic graphs. Graphs Comb. 34, 261–276 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Desormeaux, W.J., Haynes, T.W., Henning, M.A., Yeo, A.: Total domination numbers of graphs with diameter two. J. Graph Theory 75, 91–103 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dorbec, P., Henning, M.A., Rall, D.F.: On the upper total domination number of cartesian products of graphs. J. Comb. Optim. 16, 68–80 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and Intractability, A Guide to the Theory of NP Completeness. W H Freeman and Company, San Francisco (1979)zbMATHGoogle Scholar
  9. 9.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: Advanced Topics. Marcel Dekker, New York (1998)zbMATHGoogle Scholar
  10. 10.
    Henning, M.A.: A survey of selected recent results on total domination in graphs. Discrete Math. 309(1), 32–63 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Henning, M.A., Yeo, A.: A new lower bound for the total domination number in graphs proving a graffiti conjecture. Discrete Appl. Math. 173, 45–52 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Molnár, F., Sreenivasan, S., Szymanski, B.K., Korniss, G.: Minimum dominating sets in scale-free network ensembles. Sci. Rep. 3, 1736 (2013)CrossRefGoogle Scholar
  13. 13.
    Wuchty, S.: Controllability in protein interaction networks. Proc. Natl. Acad. Sci. USA PNAS 111(19), 7156–7160 (2014)CrossRefGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Computing Science and TechnologyGuangzhou UniversityGuangzhouChina
  2. 2.School of Computer Science and TechnologyDalian University of TechnologyDalianChina
  3. 3.College of Network SecurityChengdu University of TechnologyChengduChina

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